turbulent boundary layer to single-stream shear …...turbulent boundary layer to single-stream...

J. Fluid Mech. (2003), vol. 494, pp. 187–221. c© 2003 Cambridge University Press

DOI: 10.1017/S0022112003006049 Printed in the United Kingdom

187

Turbulent boundary layer to single-stream shearlayer: the transition region

By SCOTT C. MORRIS† AND JOHN F. FOSSDepartment of Mechanical Engineering, Michigan State University, East Lansing, MI 48823, USA

(Received 7 February 2003 and in revised form 16 June 2003)

This communication presents the results and conclusions of an experimental study ofthe near-separation region of a single-stream shear layer. The momentum thicknessat separation (x = 0) was θ0 = 9.6 mm, with Reynolds number Reθ =4650. Boundarylayer separation was caused by a sharp 90◦ edge. Detailed single- and multi-point measurements of the velocity field were acquired at the streamwise locations0 < x/θ0 < 100. This represents the transition region between two of the canonicalturbulent shear flows: the zero-pressure-gradient turbulent boundary layer and thesingle-stream shear layer. From the viewpoint of a separating boundary layer, theresults describe how the turbulent flow reacts to a sudden change in wall boundaryconditions. From the viewpoint of the developed shear layer, the results describethe transition to the self-similar region. The data acquired suggest that the initialshear layer instability occurs in the region very near separation (x ≈ θ0), and thatit involves only the vorticity filaments which originate in the near-wall region ofthe upstream boundary layer. This ‘near-wall region’ roughly defines the origin ofa narrow wedge-shaped domain that was identified from the velocity statistics. Thisdomain is termed the ‘sub-shear layer’. The statistics of the velocity field in the regionbounded by the sub-shear layer and the free-stream flow were found to representthe normative continuation of the upstream boundary layer. The sub-shear layerhas been found to exhibit many of the standard features observed in fully developedshear layers. For example, velocity measurements on the entrainment side of the shearlayer indicate that large-scale motions with spanwise coherence were observed. Thestreamwise dependence of the dominant frequency, convection velocity, and spanwisevelocity correlation have been documented in order to characterize the sub-shearlayer phenomenon.

1. IntroductionA single-stream shear layer is of interest as one of the canonical turbulent shear

flows and it is also important in technological applications. The region near theseparation edge of the shear layer can serve as the archetype flow for a backward-facing step, a jet exhausting from a plane surface, and other flows characterized by theinteraction between a separating primary flow and the induced secondary or entrainedflow. This communication presents the results of an experimental investigation of thenear-separation region with a turbulent boundary layer at separation. A schematic

† Author to whom correspondence should be addressed; Present address: Department ofAerospace and Mechanical Engineering, University of Notre Dame, Notre Dame, IN 46556, USA;[email protected]

188 S. C. Morris and J. F. Foss

1 m1 m1 m

δ = 100 mmδ = 100 mmδ = 100 mm

U0 = 7.1 m s–1

θ = 96 mmν/u

τ = 0.05 mm

y

x

9.75 m

ve( y = –∞) = 0.035U0

Figure 1. Schematic representation of near-separation region.

representation of the area of interest and the adopted coordinate system is shown infigure 1. Measurements were acquired in the region 0<x/θ0 < 100, where θ0 representsthe momentum thickness of the boundary layer at x = 0. This study is distinctive withrespect to the existing literature because of the relatively high Reynolds number(based on θ0) of the separating boundary layer (Re = 4650), and because the largeθ0 value (9.6 mm) permitted detailed measurements to be acquired in the region verynear separation.

The domain of interest can be viewed as the transition region between two well-studied canonical flow fields: a flat-plate turbulent boundary layer and a single-streamshear layer. This leads to two possible perspectives from which to view the acquireddata. The first perspective is to view the region shown in figure 1 as a boundarylayer with a sudden change in the wall boundary conditions. The data then serveto answer the question: how does a turbulent boundary layer react to this abrupt‘removal’ of the wall boundary condition? A related question was addressed using amoving-wall geometry in Hamelin & Alving (1996). It will be demonstrated in thefollowing sections of this paper (in agreement with the results of Hamelin & Alving)that the ‘outer region’ of the separated boundary layer remains statistically identicalto the upstream (canonical) boundary layer for several integral length scales in thestreamwise direction. The insensitivity of the outer region of the boundary layerto the sudden ‘removal’ of the bounding wall is note-worthy, and this insensitivityis expected to be of interest to researchers who study boundary layers as well asseparated flows.

Turbulent boundary layer to single-stream shear layer 189

In the second viewpoint the near-separation region is considered to be thedeveloping region of the shear layer. The data then answer the question: how does aself-similar single-stream shear layer develop from its parent boundary layer? Implicitin the concept of self-similarity is the idea that the initial conditions are ‘forgotten’(Townsend 1975). The near-separation region has been extensively studied for the caseof laminar boundary layers at separation. For example, Ho & Huerre (1984) providea thorough review and a substantial list of both single-stream and two-stream shearlayer references. Many of these studies examine the near-separation region using linearstability theory. The results of the present work will be compared and contrasted tothis established theory.

There have been a number of investigations which have looked at shear layers withvarious inflow conditions. See, for example, Bradshaw (1966), Foss (1977), Hussain &Zedan (1978), Slessor, Bond & Dimotakis (1998), Browand & Latigo (1979), andDziomba & Fiedler (1985). These studies were conducted with laminar and turbulent(tripped) separating boundary layers with Reynolds number (Reθ ) less than 1000.These references report experiments in which the upstream boundary condition ismodified and the ‘effect’ of this modification on the developed downstream flow isdocumented. A common characteristic of these studies is that the near-separationregion was not described in detail.

In summary, the present research effort serves to fill a gap in the presentunderstanding regarding the physical mechanisms which take place when a single-stream shear layer originates from a relatively high-Reynolds-number boundarylayer. The experimental programme was divided into four parts. The first wasa characterization of the boundary layer at the separation point (x = 0). Theseexperiments utilized single- and multiple-sensor hot-wire probes to document manyof the single-point statistics of the velocity and vorticity fields. These data serve todocument the inflow boundary conditions to the control volume of interest. Theresults of the second group of experiments document the single-point statistics of thevelocity in the region 0<x/θ0 < 100. In the third group of experiments, single- andmulti-point measurements of velocity were used to document many of the featuresof the unsteady irrotational flow on the low-speed entrained side of the shear layer.In the fourth part, a rake of eight hot-wire sensors was used to document thespanwise (z) correlations of the velocity field at various (x, y) locations. The resultsof the four groups of experiments will be described following an overview of theexperimental methods and the single-stream shear layer facility. Note that the resultsfrom the experiments will be presented with limited discussion. The paper ends witha list of conclusions and with a more detailed discussion of the results. This format ofpresentation was chosen so that the discussion could be provided from the perspectiveof incorporating all of the experimental data.

2. Experimental methods and apparatus2.1. Single-stream shear layer facility

A large single-stream shear layer facility at the Turbulent Shear Flows Laboratoryat Michigan State University was used for the present experiments. A schematicrepresentation is shown in figure 2. Note that the height ‘out of the page’ of the entireflow system was 1.94 m.

A constant-speed blower (not shown) moves air from the laboratory to thepoint labelled Primary inflow. The air then passes through the flow conditioners.These consist of a 2m wide honeycomb (3 mm cell × 25 mm length) section and three


Primary inflow

Flowconditioners

Contraction

Gap for contractionboundary layer removal

U0

5.74 m

Boundary layerplate

Flowconditioners

Separation edge

Entrainmentfans

9.75 m

y

x

U0

Tunnel exit

Bypass holes

Figure 2. Schematic of the single-stream shear layer facility. Note that view is rotated 90◦

with respect to figure 1.

turbulence control screens with 8 wires/cm. A contraction is located 1.2 m downstreamof the screens. This reduces the width of the flow area to 1.1 m. The boundary layerplate begins at the end of the contraction. A small gap (0.12 m) between the end ofthe contraction and the leading edge of the boundary layer plate allows the removalof the contraction boundary layer. This point of the tunnel and all downstreamlocations are at nearly zero gauge pressure with respect to the laboratory ambient.The primary flow moves through the 0.98 m wide boundary layer development section.The boundary layer plate is 5.74 m long; the free-stream velocity was 7.1 m s−1. Theboundary layer plate was laminated with 1 mm thick PVC sheeting to provide asmooth boundary condition. The boundary layer plate terminated at the separationpoint which was created using a machined 90◦ edge. The free-stream velocity was


measured during all experiments by monitoring the differential pressure drop acrossthe planar contraction. This pressure difference was acquired during all experimentsand it was found to vary by less than 0.5% over the duration of the experimentalprogramme.

The shear layer test section begins at the separation edge. The primary flow movesthrough the 9.75 m test section length with constant velocity (U0). An entrainment flowstream was directed perpendicular to the primary flow stream on the low-speed sideof the shear layer. This flow was created using four 1.2 m diameter axial propeller-typefans with an AC speed controller. This fan type is designed to move large volumes ofair at relatively low pressure rise (∼25 Pa). For this reason, bypass holes were createdsuch that the fans would operate at design conditions to prevent unsteadiness fromblade stall. That is, most of the air moved by the fans is recirculated back to thelaboratory. The ‘plenum’ between the fans and the flow conditioning is maintainedat a pressure which is slightly higher than the atmospheric value. The entrainmentflow conditioning is identical to the primary flow conditioning. The lateral location ofthe entrainment screen closest to the shear layer is y = −2.75 m, using the coordinatesystem shown with origin at the separation edge.

The entrainment fans were adjusted to provide the correct entrainment rate fora shear layer at zero pressure gradient. This was verified using hot-wire sensorsin the primary flow at several streamwise locations. That is, the fans were adjusted suchthat ∂U0/∂x ≈ 0. The resulting entrainment velocity was found to be ve = 0.035U0 ≈0.25 m s−1.

The exit of the tunnel was open to the laboratory. The nearest obstruction to theoutflow was a building wall located 6.1 m from the tunnel exit. The ceiling and floorof the laboratory were respectively located approximately 2 m above and below theshear layer test section. These conditions provided sufficient space to turn the exitingflow without disturbing the shear layer at the end of the test section.

2.2. Instrumentation

Time series of the fluid velocity were measured using single- and multi-sensor hot-wireprobes. The sensor elements were made from 5 µm diameter tungsten wire. The totalwire length was 3 mm, with a 1 mm region on each side that was plated with 50 µm ofcopper to support the 1mm active portion of the wire. The sensors were controlledusing DISA 55M anemometers with a heating ratio of 1.7. The frequency responsewas always greater than 20 kHz. All sensors were calibrated before (pre) and after(post) each experiment with a maximum of three hours between calibrations. Unlessthe pre and post calibrations agreed to 2% or better the data were retaken.

Measurements acquired in the separating boundary layer also include two-component (u, v) data acquired using an X-wire probe. Additionally, vorticity timeseries data were acquired using a compact four-wire probe. A full description of theseprobes and processing algorithms can be found in Foss & Haw (1990), Wallace &Foss (1995), and Morris (2002). A thorough review of hot-wire techniques is given byBruun (1985).

3. Experiment 1: characterization of the boundary layerThe study of the near-separation region began with detailed measurements that were

acquired very near to the separation point (x ≈ 0). These data and their interpretationserve to document the inflow boundary conditions to the separated flow discussedin the following sections. The investigation of the boundary layer was divided into


two experiments. First, measurements to identify possible effects of separation onthe boundary layer upstream of the separation edge were executed. Second, pointstatistics in the boundary layer were measured with enough detail to ensure that anearly canonical zero-pressure-gradient boundary layer existed and that it comparedwell with standard measurements as reported in the literature.

3.1. Shear layer to boundary layer ‘communication’

The purpose of this experiment was to answer the question: do the pressurefluctuations from the shear layer significantly alter the characteristics of the boundarylayer at separation?. In other words, do the large-scale spanwise coherent motionsin the shear layer cause a measurable change in the boundary layer characteristicsupstream of separation? The answer to this question for the condition of a laminarboundary layer at separation is discussed in Holmes, Lumley & Berkooz (1992). Theirresults suggest that the shear layer does influence the separating boundary layer forlaminar separation. In contrast, it will be argued from the present data that this isnot true for the present (high-Reynolds-number turbulent) boundary layer.

It is known (see e.g. Browand & Trout 1980) that the velocity field in the irrotationalfluid near a planar shear layer is unsteady. This is a result of the large-scale motionswhich exist within the turbulent region. If these motions also affect the boundarylayer flow before separation, it is intuitively reasonable that a non-zero correlationbetween velocities in the boundary layer and in the entrainment stream would provideevidence that the shear layer was affecting the boundary layer. It is noted, however,that statistical quantities such as linear correlations do not indicate causation, andshould only be used as an indicator that a mechanistic coupling exists between thetwo regions.

Simultaneous hot-wire measurements were taken in both the entrainment streamand the boundary layer flows to test this hypothesis. A single hot-wire sensor wastraversed in the entrainment stream near the separation point with the sensor axisaligned with the z-direction such that the components of velocity in the (x, y)-plane would be recovered. Note that the w component should be quite small in theessentially two-dimensional entrainment region. The entrainment probe was traversedthrough a range of locations in the region 0 <x/θ0 < 20, y < 0. An X-wire probe wassimultaneously placed in the boundary layer in the region 0 <y/θ0 <10, −4 <x/θ0 < 0,such that the u and v components of velocity were recovered. Time series data fromthe X-wire and the entrainment probe were recorded for 40 seconds at 5000 Hz. Thestandard normalized cross-correlation as a function of τn, the discrete time delayvariable, is defined as

R(τn) =v1(tn)v2(tn + τn)

v1v2

. (1)

The variable v1 is the u or v component of velocity measured from the X-wire, v2 isone of the signals from the entrainment probes, and the overbar represents the timeaverage. The tilde represents the RMS value of that signal.

The results of these calculations have shown nearly zero correlation values for allof the above measurement locations. That is, there was no measurable correlationbetween the two signals for any of the spatial locations measured. A second calculationwas performed in which a low-pass filter was used to allow only frequencies lessthan, for example, fc =100 Hz (where fcθ(0)/U0 = 0.135) to be correlated. Thistype of filtered cross-correlation has been shown to be effective in detecting non-zero correlations in highly turbulent flow fields which contain broad-band spectral


x/θ0 = +0.1

x/θ0 = –0.1

0

5

10

15

20

25

30

q–+

1 1 0 100 1000

0000 5 10

2

4

6

8

10

15–5–10–15

q+

y+

y+

u+ = 2.31n( y+)+6.7

u+=2.51n( y+)+5.5

u+= y+

Figure 3. Boundary layer mean velocity. The y+ =0 point (+) on the inset figure was addedfor clarity, based upon the no-slip condition. Note that U0/uτ = 25.5.

information; see e.g. Naguib (1992). These cross-correlations, however, were alsofound to be negligible in magnitude for all of the acquired data.

It is inferred from the above observations that the upstream ‘communication’ isnot measurable in a separating turbulent boundary layer. It is therefore reasonableto assume that the measurable characteristics of a high-Reynolds-number turbulentboundary layer upstream from a fixed separation point with zero pressure gradientare identical to those which would exist if the separation were not present. Theseconclusions are further supported by the results of the following section, whichpresents detailed velocity measurements for x ≈ 0. As indicated earlier, this result isin contrast to observations described in Holmes et al. (1992) for a separating laminarboundary layer. Further discussion will be given in the conclusions section.

3.2. Boundary layer statistics near x = 0

The single-point statistics of the boundary layer in the region very near separationwere measured using a single hot-wire sensor. The probe was calibrated and traversedthrough the boundary layer at the streamwise locations x/θ0 = −0.1 and +0.1. Eachdata point was acquired at a sampling frequency of 5000 Hz for t = 30 s, wheretU0/θ0 = 2.2 × 104. The sensor was aligned in the z-direction, such that the magnitudeof the velocity in the (x, y)-plane was recovered:

q(t) =√

u2 + v2. (2)

Most authors do not make the distinction between u and q in single-probe hot-wiremeasurements. This is because the v component is typically small compared to u inwall-bounded flows. The emphasis on q in the present work is important because thesurveyed region (x > 0, y < 0), will be dominated by the v component of velocity. Incontrast, for the region (x ≈ 0, y > 0) the approximation q ≈ u is nearly exact.

The time average and standard deviation of the velocity measurements will bedenoted by q and q respectively. The profiles q(x = ±0.1θ0, y) are shown in figure 3.Note that these data are normalized by the viscous units (uτ =

√τw/ρ for velocity and


x/θ0 = +0.1

x/θ0 = –0.1

0

0.5

1.0

1.5

2.0

2.5

3.0

q~+

1 1 0 100 1000

00 5

1

2

3

–5–10

y+

y+

u~+

10 15 20 25 30

Figure 4. Boundary layer velocity RMS.

ν/uτ for y, where τw is the time-averaged wall shear stress, ρ density and ν viscosity)in log–linear coordinates. The wall shear stress was computed from the slope of alinear fit of the data points closest to the wall at x/θ0 = −0.1. Note that the closestlocation to the wall is y+ ≈ 3.2, or roughly y ≈ 150 µm, which is far enough from thewall to ignore proximity effects in the hot-wire readings. A typical log–linear regioncan be observed through the range 40 <y+ < 800. This region roughly approximatesthe standard fit q+ =2.5 ln(y+) + 5.5. The present data are best represented byq+ =2.3 ln(y+) + 6.7 for 30 <y+ < 500.

It is of interest to compare the two velocity profiles which have been acquired. Themeasured x/θ0 = ±0.1 mean values agree for all locations where y+ > 5 (y > 0.25 mm).The inset to the figure shows the deviation of the data from the two traverses fory+ < 5 in linear coordinates. The deviation between the two profiles indicates a strongstreamwise gradient in velocity given that the x displacement of the sensor for thetwo traverses was 0.19θ0 (1.8 mm). For example, �q/�x ≈ 400 (s−1) at y = 0.

The integral statistics of interest, which were calculated from the mean profile, werethe momentum thickness θ = 9.62 mm, and the displacement thickness δ∗ = 12.6 mm.The Reynolds number based on momentum thickness was Reθ = 4650. Additionalparameters are the friction coefficient and the shape factor, which are defined asCf =2(uτ/U0)

2 and H = δ∗/θ , respectively. These were determined to be Cf = 0.00295and H = 1.31.

The q(x = ± 0.1θ0, y) profiles in viscous units are shown in figure 4. It is observedthat the majority of the y+ > 0 region is invariant in the q data, as observed in figure 3.Specifically, q is invariant for y+ > 30, whereas q is invariant for y+ > 5. Note thatthe maximum value for the x/θ = −0.1 data is q/uτ = 2.74. Other researchers (forexample Klewicki & Falco 1990) have found the peak in the fluctuation intensityto be ∼2.9 for similar Reynolds numbers. This discrepancy can be explained bythe relatively large sensor length used in the present study. Specifically, the hot-wirelength in viscous units was λ+ =20ν/uτ , whereas Klewicki & Falco (1990) recommend


0

0.5

|ω+|

10 100 1000

010010

2

4

y+

y+

3

≈10 θ

0.4

0.3

0.2

0.1

–0.1

0.7

0.6

1

|ω|ω~

Figure 5. Mean and RMS of vorticity at separation. –◦–, mean value from slope of singlesensor traverse. Error bars denote ±1 standard deviation of vorticity calculated from thefour-wire vorticity probe time series.

a sensor length of less than 10v/uτ in order to avoid spatial averaging of the velocityfluctuations.

The spanwise vorticity in the boundary layer is also a variable of interest.The magnitude of the time-averaged vorticity in viscous units is approximatedby |ω+

z | = ∂q+/∂y+. The slope of the mean velocity profile shown in figure 3 wascomputed; see figure 5. The standard deviations of the fluctuations in spanwisevorticity, ωz, were measured using the four-wire vorticity probe; these data are alsoshown on figure 5. Note that the spatial resolution of the probe is established by thedistance between the two straight wires. This is nominally 1 mm (20 viscous wall unitsin the present study). The measured values of ωz are consistent with the data presentedin Klewicki, Falco & Foss (1992) and references therein. These data show thatboth positive and negative values of vorticity were realized for y+ > 20. This can becompared with the near-wall region wherein only one sign of vorticity that is consistentwith that of the mean shear is observed. The vorticity fluctuations normalized by thelocal mean value: ωz/|ωz| have been plotted for the region 10 <y+ < 800 as the insetto figure 5. Note that the values of ωz/|ωz| are uncertain for y+ > 800 because boththe numerator and denominator are approaching zero. The data show that the relativefluctuation levels of vorticity increase approximately exponentially (linear in semi-logcoordinates) throughout the ‘log region’ (10 < y+ < 600) of the boundary layer. Thecomparison of the two representations of ωz in figure 5 are instructive in showingthat the absolute fluctuation levels of vorticity decrease with increasing y+, yet thevalues are increasing with respect to the local mean vorticity.

Qualitative aspects of the vorticity filament topology can be inferred from thesemeasurements and the solenoidal condition of the vorticity vector field. At a smalldistance from the wall the filaments are necessarily perpendicular to the local shear


0 1 2 3 4 5

x/θ

0

1

2

3

–1

y–θ

Figure 6. Sample flow visualization of the near-separation region.

stress vector, and parallel to the surface. The observation that ωz(t) < 0 in the near-wallregion implies that the near-wall filament topology is relatively simple. In contrast,the measurements farther from the wall (y+ ∼ 20) indicate that the vorticity filamentsmust be sufficiently complex or ‘twisted’ such that vorticity values of both sign arerealized at a given y location. Farther from the wall, the statistics of the velocity andvorticity fields exhibit many of the properties of isotropic turbulence; see, for example,Saddoughi & Veeravalli (1994) for examples of velocity statistics which support localisotropy in the log region of boundary layers. These features of the vorticity field willbe considered in the following sections which describe the near-separation region ofthe boundary layer.

4. Experiment 2: point statistics and the ‘sub-shear layer’A flow visualization image of the near-separation region is shown in figure 6. This

digital image was acquired using a Kodak 1k × 1k camera and a nominally 1 mmthick laser light sheet oriented in the (x, y)-plane. The boundary layer flow was seededusing an oil-fog system located upstream of the boundary layer leading edge. Theunmarked entrainment flow provided the contrast between the two fluid streams.Specifically, the marked primary fluid appears white, and the clean entrainment fluidappears dark in the figure. Note the scale and orientation with respect to figure 1.Specifically, recall that the boundary layer thickness δ is of order 10θ . This pictureis not repeatable in detail because the flow is turbulent in this region. However, thegeneral features represented and described are common to all realizations.

Several interesting observations can be made from this visualization. First, astreakline from the separation point can be easily identified. This shows a ‘wavy’or ‘flag-flapping’ type of behaviour. The first disturbance to the streakline occurs at astreamwise location of x ≈ θ0. The ‘size’ of this motion appears to be of order 0.5θ0.This observation is consistent with the ‘communication’ results described previously.


Free stream

Canonical turbulent boundary layer

Sub-shear layer

Developing shear layer

Significant irrotational motions

Entrainment stream

Figure 7. Diagram of flow regions; not to scale.

That is, it would seem unlikely that the pressure fluctuations from this first disturbanceor the downstream disturbances would influence the upstream boundary layer giventheir relatively small scale. Note also that the asymmetry of the shear layer is evidentfrom the movement of the marked fluid into the entrainment region (y < 0), whereasvery little clean fluid is observed for y > 0. The visually apparent ‘growth’ of the shearlayer is clearly at a length scale which is significantly smaller than the boundary layerthickness.

Based on these observations, and equipped with the data to be presented inthis and subsequent sections, several distinct regions of the flow field have beenidentified. These are shown schematically in figure 7. These regions and their definingcharacteristics will provide a conceptual framework and vocabulary which will beuseful in describing the results to follow. The canonical turbulent boundary layer is theregion which exhibits stochastic values of the velocity field that reflect the evolutionof a boundary layer that is still attached to a flat plate. The sub-shear layer representsthe turbulent region where the flow is measurably different from that if the boundingwall were present. The region marked significant irrotational fluctuations representsthe domain where measurements of the velocity fluctuations exhibit essentially nohigh-frequency (small-scale) motions. It is inferred that the unsteady aspects of thesemotions are driven only by the unsteady pressure field created by the shear layer.Lastly, the transition between the sub-shear layer and the developing shear layer arenot well defined. Hence no such boundaries are drawn in figure 7.

Point statistics were acquired using a single hot-wire probe which was traversedacross the shear layer at 12 streamwise locations. The data were acquired at 5 kHz for60 s. If the characteristic velocity is taken as U0/2, and if the shear layer width is takento be 6θ(x), then the 60 s represents 500 to 2000 shear layer widths for the 12 locations.The sensor was aligned in the z-direction to recover the time series of q as describedin the previous section. Note that in the high-speed free stream the sensor respondsto q = u =U0, and in the entrainment stream the sensor responds to q = v = ve. Thetime-average velocity (q) and the standard deviation of the fluctuations (q) were thencomputed from the time series values. The mean velocity is shown in figure 8 as an‘x, y plot’, and in figure 9 as a contour plot. The values of q are shown in figures 10and 11 in x, y and contour formats, respectively. Note that the boundary of thesub-shear layer region, identified qualitatively in figure 7, can be roughly describedby the q/U0 = 0.087 contour in figure 11.

A striking feature observed in these data is the nearly streamwise-invariantboundary layer statistics. That is, the gradients ∂( )/∂x that are known to be smallin boundary layers are also very small downstream of separation. As a specific


0

0.2

0.4

0.6

0.8

1.0

q–

U0

0 5 10 15 20 25–5–10–15–20–25

0 1 2 3–1–3 –2

0.4

0.2

0.6

0.8

y/θ0

101

84.4

68.8

54.2

40.6

29.6

19.3

12.0

7.19

3.54

1.88

0.73

–0.1

x/θ0

Figure 8. Mean velocity profiles for 0 < x/θ0 < 100.

0 5 0 100

0

20

–20

x/θ0

θ0

y

Figure 9. Contour plot of mean velocity, q/U0. The grey ‘grid’ represents the (x, y) locationswhere data were collected.

example, the mean profiles are identical to the boundary layer profile for y/θ0 > 2in the range 0 < x/θ0 < 29. The profiles of q are streamwise invariant for y/θ0 > 4 inthe range 0 <x/θ0 < 29. These features are shown by the nearly streamwise-invariantcontours in figures 9 and 11, or by close examination of the individual profiles infigures 8 and 10. Several histograms of velocity are given in figure 12 for the points:(x/θ0, y/θ0) = (0.73, 2), (7.2, 2) and (19, 2). Although differences can be noted, thesimilarity of these histograms provides a specific example of the nominal streamwiseinvariance of the moments of the velocity field.

It is noteworthy that these results are consistent with those of Hamelin & Alving(1996). In that reference, a moving belt was used to remove the inner layer of anReθ = 2800 turbulent boundary layer in order to study the stochastic evolution of


0

0.02

0.04

0.06

0.08

0 10 20 30–10–20–30

0 1 2 3–1–3 –20.05

y/θ(x = 0)

101

84.4

68.8

54.2

40.6

29.6

19.3

12.0

7.19

3.54

1.88

0.73

–0.1

x/θ(x = 0)

0.10

0.12

0.14

q~

U0

0.07

0.09

0.11

0.13

Figure 10. RMS of velocity for 0 < x/θ0 < 100.

0 5 0 100

0

20

–20

x/θ0

θ0

y

Figure 11. Contour plot of velocity RMS q/U0.

the outer scales. The present work is similar in that the streamwise evolution of theboundary layer is measured after an abrupt change in wall boundary conditions. Inthe case of the moving-wall boundary condition, the mean flow field was streamwiseinvariant for y/θ0 > 2 in the range 0 <x/θ0 < 34 (see figure 5 of Hamelin & Alving).The RMS of the velocity fluctuations at a given y location was streamwise invariantfor y/θ0 > 4 in the range 0 <x/θ0 < 30 (figure 7 of Hamelin & Alving). These regions ofstreamwise invariance are nearly identical to those realized in the present experiment.

Two variables of interest were calculated from the mean velocity profiles: themomentum thickness of the shear layer θ(x) and the maximum slope of the meanvelocity. The calculation of θ requires the integration of the streamwise component ofvelocity across the shear layer. Because the magnitude of the velocity was measuredas opposed to strictly the u component (see equation (2)), an arbitrary cut-off ofq/U0 = 0.05 was used to end the integration on the low-speed side. This is a common


0

0.08

0.16

0.04

0.12Fr

acti

on o

f re

aliz

atio

ns

0.50 0.75 1.00

q/U0

x/θ0= 0.73, y/θ0= 2x/θ0= 7.2, y/θ= 2

x/θ0=19, y/θ0= 2

Figure 12. Streamwise evolution of velocity histogram at constant y/θ0 = 2.0.

0 20 40 60 80 100 120

x/θ0

θ ~ 0.035x

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

θ0

θ

Figure 13. Momentum thickness vs. x/θ0.

technique used by, for example, Ali et al. (1985) and Hussain & Zaman (1985).These results are shown in figure 13. The growth rate in the region x/θ0 > 60 wasfound to be 0.035 as measured previously by Foss (1994). Note that Hussain &Zaman (1985) found a linear growth rate of 0.032. The second variable derived fromthe point statistics that will be of interest in subsequent sections is the maximumslope of the velocity profile. These data are shown in log–log format in figure 14for streamwise locations (1.8 < x/θ0 < 550). This extended domain shows that the


11

10 100

10

1000

100

~1/x0.79

x/θ0

dqdy max

Figure 14. Maximum velocity gradient vs. x/θ0.

0.120.120.12

0 50 1000.11

0.14

0.130.130.13

0.15

0.17

0.16

150 200 250 300 350

x/θ0

q~max

U0

0 5 10 15 200.110

0.115

0.120

0.125

0.130

Figure 15. Peak value of velocity RMS as a function of streamwise location.

function (dq/dy)max ∼ 1/x0.79 agrees reasonably well throughout the sub-shear layerregion as well as farther downstream.

A variable of interest is the maximum value of q obtained at a given streamwiselocation. This information was obtained from the data shown in figure 10, and plottedin figure 15. The most interesting feature of these data is that the local maximumvalue is realized at x/θ0 ≈ 2. This peak occurs at a lateral position of y/θ0 = 0.15.Previously, the existence of a local maximum has only been observed in shear layerswith a laminar separating boundary layer, whereas the tripped or high-Reynolds-number boundary layers lead to a monotonic increase in q; see Foss (1977) andHussain & Zedan (1978). The present data support the concept that the sub-shear


layer region very near the separation point is a viscously dominated flow region whichshares many of the characteristics of low-Reynolds-number separations. It is inferredthat this detailed feature of the flow field near the separation edge was previously notobserved given the small dimensions of θ0 in those experiments.

As a final note regarding the sub-shear layer, the observations made also provideinsight into the transition length required in order to obtain a self-similar shear layer.Specifically, it is known that shear layers which originate from a laminar boundarylayer require a streamwise development of 1000θ0 in order to be self-similar (Bradshaw1966). In contrast, self-similarity is observed in shear layers which originate from aturbulent boundary layer at streamwise locations less than 150θ0, as observed in thepresent experiments, as well as in the literature (see e.g. Hussain & Zaman 1985).Dimotakis & Brown (1976) suggested that the streamwise distance to self-similarity isrelated to the wavelength of the initial stability (λ0), and not to the initial momentumthickness (θ0). They also indicate that the case of turbulent separation will probablyhave shorter length scales compared to θ0. This can be expressed as the ratio λ0/θ0.This value has been found to be λ0/θ0 ≈ 10 for laminar separation, and λ0/θ0 ≈ 1 inthe present study (see figure 6). The present result clearly supports the inference madeby Dimotakis & Brown. Further discussion will be given in the conclusions section.

5. Experiment 3: shear layer diagnostics from entrainment region measurementsA feature of the single-stream shear layer is the ubiquitous large-scale coherent

motions. These motions are characterized by a correlation length in the spanwise(homogeneous) direction that is many times larger than the local shear layer thickness.The existence, physical makeup, and dynamic significance of these large scales arestill topics of debate and research. The origin of these motions is believed to be theinflectional instability of the shear layer velocity profile. The flow visualization andpoint statistics described in the previous section indicated that this instability doesnot influence the entire separating boundary layer. Rather, it is suggested that theinfluence of these motions is limited to the region identified as the sub-shear layer.A simple method of detecting these large-scale motions is to acquire velocity timeseries in the irrotational flow field near the shear layer. Many studies have made useof the fact that the time signature of velocity in both the high-speed and low-speedstreams can be used to identify features of the large-scale motions within the shearlayer; see for example, Browand & Weidman (1976). Koochesfahani et al. (1979),Browand & Trout (1985), and Narayanan & Hussain (1996). The efficacy of thistechnique is based upon the pressure fluctuations, which are mechanistically linkedto the large-scale motions. This leads to easily detectable fluctuations in the velocityfield of the irrotational flow which bounds the vortical shear layer.

The observations made in the entrainment stream will be described in threesubsections: the streamwise dependence of the convection velocity of the coherentmotions, the spectral properties of the entrainment stream, and the frequency andscaling of the initial instability.

5.1. Convection speed of the coherent motions

A variable of interest is the convection velocity Uc of the spanwise motions. Manyinvestigators of both single- and two-stream shear layers have found the convectionvelocity to be the mean velocity of the two streams with equal density. See forexample, Dimotakis (1986) and references therein. The ratio would be predicted to beUc/U0 = 0.5 in the single-stream shear layer. The time-averaged convection velocity


0.2

0 20 100

0.4

0.6

120 300200

x/θ0

0.8

1.0

40 60 80 400 500 600 700

2Uc

U0

Figure 16. Convection velocity of spanwise motions. The break in the x-axis is used to showthe x/θ0 < 120 region in more detail.

was measured from time series data of two hot wires placed in the entrainment streamwith a streamwise displacement �x. The lateral locations where q(y) ≈ 0.015U0 weredetermined to have significant signal amplitude in the irrotational region. The time-delayed cross-correlation of the two hot-wire signals was computed from these timeseries to locate the time delay to the first observed positive peak: �t . The averageconvection velocity is defined as Uc = �x/�t .

Figure 16 shows the normalized convection velocity 2Uc(x)/U0. Of interest is thelow value of 0.54 near x = 0 and the increase to the expected value of 1.0 forx/θ0 > 120. It is inferred from this result that only the vorticity from the near-wallregion participates in an inflectional instability in the region very near separation,and that the percentage of the vorticity (on average) that participates in the shearlayer instability increases in the x-direction. The former inference is clearly supportedby the independent visualization image of figure 6. Additional statements can bemade to support this inference. The first observation is that the velocity gradients arevery steep in the region x = y = 0 which lead to length scales which are considerablysmaller than the local integral scales. Since the length scale of the most amplifiedmotions will scale with the velocity gradient at the point of inflection, it is reasonablethat only the fluid in the y ≈ 0 region will participate in the instability.

A second argument to support this assertion is related to the vorticity field of theturbulent boundary layer upstream of separation. It is known that the wall confinesthe movement of the vorticity filaments in the near-wall region, and as a result theyare relatively organized compared to the outer portion of the boundary layer (seeKlewicki et al. 1992). Note also the large values of ωz/|ωz| in the outer region infigure 5. It can be inferred from these observations that the ‘organized’ near-wallvorticity participates in the first instability, whereas the ‘disorganized’ outer motionsdo not.

It is possible to calculate how much of the vorticity participates (on average)in the first instability based on the above observations. Given the relatively smallcontribution of the y component of velocity to the vorticity in the near-wall region,


0.1 1 10 100 1000

Frequency (Hz)

100.1

84.2

68.654.040.529.5

19.2

12.0

7.17

3.53

1.87

0.73

(log

sca

le)

x/θ0

Figure 17. Velocity spectra of entrainment for the 12 streamwise locations measured.

the mean streamwise velocity can be written as

q(y) =

∫ y

0

ω dy. (3)

As described above, the convective velocity of the first instability was found to be0.27U0 from figure 16 near x =0. The ‘scale’ of velocity in this context (2Uc = 0.54U0)can be interpreted as an ‘apparent free stream’ speed as observed by the vorticityfilaments which participate in the first instability. The location where q/U0 = 0.54 wasfound to be y+ ≈ 42 at x = 0, see figure 3. Evaluating equation (3) at y+ = 42 showsthat 54% of the time-averaged vorticity which separates from the boundary layerparticipates in the first instability. The increase in the coherent-motion convectionvelocity shown in figure 16 indicates that an increasing amount of the time-averagedvorticity from the boundary layer participates in the coherent motions as the flowevolves in the streamwise direction.

5.2. Spectral properties of the entrainment stream

The spectral properties of the entrainment stream in proximity to the active shearlayer were used to investigate the coherent motions of the sub-shear layer region. Thepower spectral density (the Fourier transform of the autocorrelation) was computedfor a single lateral (y) location at each of the 12 (x > 0) streamwise locations identifiedpreviously in figure 8. The lateral positions where q(y) ≈ 0.015U0 were selected onthe low-speed side where the velocity fluctuations appeared irrotational. These dataare shown in log–log format in figure 17. The amplitude of each data set was shiftedvertically in order to distinguish the data sets on one figure. These data show a


0 10 20 30 40 50 60 70 80

(ms)

–0.4

0

–0.2

0.2

0.4

0.6

0.8

1.0

Aut

ocor

rela

tion Peak location for

fm calculation

Figure 18. Sample of the autocorrelation function in the entrainment stream at x/θ0 = 7.17.

distinct local maximum, or ‘hump’, in the spectrum for all streamwise locations withthe exception x/θ0 = 0.73. For example, the hump at x/θ0 = 29.5 occurs at 10 Hz.Nearly identical results for the range 37 <x/θ0 < 4800 can be found in Hussain &Zaman (1985, hereafter referred to as H-Z). It can also be observed that the frequencyat which the local maximum occurs, fm, decreases with downstream distance. Thisfrequency can also be identified using the first peak of the autocorrelation function.This leads to a similar, although not necessarily the same, identification of a preferredfrequency. Figure 18 shows an example of the autocorrelation function for x/θ0 = 7.17.It should be noted that at all locations, the ‘hump’ does not imply that the signal isperiodic. These humps should be interpreted only as a preferred frequency at thatstreamwise location.

The frequency values fm (normalized by θ0 and U0) were calculated from boththe spectra and autocorrelations for the region 1.8 <x/θ0 < 650; see figure 19. Thefrequency data from H-Z in the region 37 <x/θ0 < 4800 (taken from figure 5 of thatreference) are also shown in figure 19. Note that the boundary layer Reynolds numberat separation was Reθ = 428 in H-Z. The agreement is quite good in the overlap regionbetween these data sets, which indicates that the dimensionless frequency at thesex/θ0 locations is not a strong function of the Reynolds number of the separatingboundary layer.

The frequency data can also be made non-dimensional using the free-stream velocityand the local momentum thickness. That is, the fm and the θ(x) values can becombined to form a new variable defined as f ∗ = fmθ(x)/U0. These data are shown infigure 20. The f ∗ value measured at the location x/θ0 = 600 is also shown on the figure.The functional dependence of f ∗ can best be interpreted by considering the functionaldependence of both the momentum thickness and the frequency from the respectivecurve fits of the data. Although the momentum thickness does not grow linearly inthe region very near separation, all streamwise locations shown in figure 13 can beroughly approximated by

θ(x)

θ0

= 0.0312

(x

θ0

)+ 1.079. (4)


10–4

10–3

10–2

10–1

103102101100

x/θ0

0.133(x/θ0)–0.71

Autocorrelation results

Spectra results

Hussain & Zaman (1985)

~1/x

fmθ0

U0

Figure 19. Streamwise variation in peak frequency.

0

0.04

0.08

0.12

0.16

20 40 60 80 100x /θ0

0.098

0.026

Values atx /θ0=600

fm(x) / (du–/dy)max(x)

fm(x)θ(x)/U0

Figure 20. Non-dimensional frequency vs. x/θ0.

A fit of the frequency data is represented by

fmθ0

U0

=0.133

(x/θ0)0.714. (5)

Therefore, the dimensionless frequency becomes

f ∗(x) =fmθ(x)

U0

= 0.0041(x/θ0)0.286 + 0.143(x/θ0)

−0.714. (6)

Equation (6) is shown as the solid curve on figure 20. The two terms of equation (6)represents two limiting regions of interest: the region very near separation, and thebehaviour for large x. The larger values of x/θ0 are clearly dominated by the firstterm of equation (6). This equation would predict an f ∗ which is slowly increasing for


large x/θ0 values because the momentum thickness grows linearly, and the exponentof the frequency dependence is −0.71. This is in contrast to the expected resultthat the dimensionless frequency would be constant. It is also noted that the largestdownstream distance measurement location provided f ∗(x/θ0 = 600) = 0.026. This issomewhat (25%) larger than the value of 0.021 from the x/θ0 < 100 data, althoughthis difference is close to the uncertainty of the measurement. Note also that H-Zmeasured a value of 0.024, although the uncertainty in those measurements seems tobe considerably greater given the scatter in the H-Z data.

The x dependence of f ∗ has two possible explanations: either the f ∗ values arestreamwise dependent for all x, or the shear layer is still ‘relaxing’ from the initialupstream boundary conditions (i.e. the turbulent boundary layer). The second of theseexplanations is likely to be the correct one. It can be observed from the H-Z datain figure 19 that afm ∼ 1/x dependence approximates the data reasonably well forx/θ0 > 1000, whereas fm ∼ 1/x0.71, as measured in the present data, seems to correlatewell with the H-Z data for x/θ0 < 1000. It is possible that the present shear layerwould also tend towards a 1/x dependence for larger x values given the agreementwith the H-Z data in the region where the data overlap.

The non-dimensional frequency in the limit of small x values is also of interest,particularly with regard to the sub-shear layer described in the previous sections.In terms of equation (6), the second term dominates the magnitude of f ∗ for smallx/θ0. This is because the frequency increases as x is decreased, and the momentumthickness θ(x) approaches the value at separation (θ0). A preferred scaling mightbe obtained by relating the frequency to the local maximum velocity gradient:f ∗

grad(x) = fm/(du/dy)max . Note that this definition is equivalent to using the ‘vorticitythickness’ scaling. These data are also shown on figure 20. This definition wasmotivated by the fact that both variables used in the definition of f ∗

grad(x) follow apower law ∼x−n as shown in figures 14 and 19. It can also be observed that theexponent of the maximum velocity gradient, n= 0.79, closely matches the value 0.71measured for the preferred frequency.

5.3. The initial instability

The frequency of the initial instability has been the focus of numerous studies (seee.g. Ho & Huerre 1984). The understanding of how the boundary layer vorticity isredistributed downstream of separation is of considerable interest both fundamentallyand technologically. Manipulation and periodic forcing of the boundary layer atseparation can take particular advantage of the first instability in order to controlthe unsteady dynamics of the shear layer.

The near-separation region of a laminar shear layer is known to be unstableto small-amplitude perturbations as described by the Kelvin–Helmholtz instability.Linear stability theory has been used by many authors to understand and predictthe frequency and growth rate of the first instability. There are several importantassumptions of the linear theory which often include parallel flow, laminar flow withinfinitesimal disturbances, and a specific velocity profile such as a hyperbolic-tangentfunction. Although these assumptions are not physically realized in any turbulentshear flow, the theory predicts the frequency of the fundamental instability for variousgeometries with marked success. The efficacy of this theory for shear layers and jetswas reviewed by Ho & Huerre (1984) and Thomas (1991). The Strouhal number ofthe most amplified frequency is defined as St0 = fmθ0/U0. Note that some referencesuse the average velocity in the case of a two-stream shear layer. This would change


0

024

–20.02 0.04 0.06 0.08 0.10

Time (s)

(a)

(b)

(c)

(d )

024

–2

–2

0

2

–2

0

2

q�(t)

q~

Figure 21. Time series samples from x/θ0 = 0.73.

the definition by a factor of two; the referenced values have been divided by twowhere appropriate for comparison to single-stream results.

In contrast, the effects of a turbulent boundary layer state and the Reynoldsnumber on the initial instability have not been extensively investigated. Although theinitial development region for laminar flow has received considerable attention, mostshear layer studies which have a turbulent boundary layer do not consider the initialinstability of the shear layer. Ho & Huerre (1984) comment that the most amplifiedStrouhal number changes from 0.016 for a laminar flow to 0.022–0.024 for turbulentflow as a result of ‘presently unexplained reasons’. The remainder of this section willclarify their observation in terms of the sub-shear layer phenomenon.

The single-sensor hot-wire data described in the previous sections were used toobserve the frequency of the initial instability. Using the spectral information shownin figure 17 it can be observed that the time-averaged spectra of velocity show a localmaximum at the streamwise locations x/θ0 � 1.87. Although these time serics arenot periodic, the local maximum is interpreted to represent the preferred frequency.However, the lack of a local maximum in the spectra for x/θ0 = 0.73 does not implythat a preferred frequency does not exist. The flow visualization shown in figure 6indicates that the first instability occurs in this region. This has motivated a moredetailed examination of the time series data at this location to identify if a preferredfrequency can be identified.

Several segments of time series data are shown in figure 21. These data show thatpseudo-periodic motions do exist, although the frequency, phase and amplitude varyconsiderably during a relatively short time period. This explains why the spectrafailed to show a local maximum value at a particular frequency. This result is alsophysically reasonable given that the vorticity convected from the viscous region ofthe boundary layer is highly unsteady and highly perturbed by the turbulence in the‘log region’. An alternative approach to identifying a preferred frequency was taken,using only short segments of the times series. Specifically, the time series data weredivided into 0.1 and 0.2 segments for analysis. The autocorrelation was calculated foreach of these segments. Several examples of these correlations are shown in figure 22along with the autocorrelation of the full time record. The advantage of using theshort time autocorrelations is that a distinct local maximum occurred in many of


1.0

0

0.2

0.4

0.6

0.8

–0.20 0.005 0.010 0.015 0.020

Time delay; τ (s)

(a)

(b)

(c)

(d )

Figure 22. Autocorrelations of the time series at x/θ0 = 0.73. The long time averaged(thick line) as well as 0.1 s averages from the time series in figure 21 are shown.

060

0.05

0.10

0.15

80 100 120 140 160 180 200 220

0.1 s samples

0.2 s samples

Frequency of local maximum (Hz)

Frac

tion

of

real

izat

ions

Figure 23. Histogram of observed frequencies from short time autocorrelations.

the calculations, indicating a preferred frequency during that time segment. Localmaxima were found in 51% of the 0.1 s time series and 37% of the 0.2 s time series.A histogram of the observed frequencies is shown in figure 23. The mean frequencyfrom the 0.1 s sampled was 130 Hz. The mean value from the 0.2 s samples was146 Hz. These values are in nominal agreement with the value of 124 Hz obtained byextrapolating the curve fit given by equation (5). This agreement adds confidence thatthe coherent motions observed in the flow visualization at the location x/θ0 = 0.73 doexhibit a preferred frequency.

The next step in the determination of St0 is to choose velocity and length scales tonon-dimensionalize the indicated frequency. The standard choice of U0 and θ0 wouldnot be appropriate since it is only the near-wall region that is participating in this


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

–0.6 –0.4 –0.2 0 0.2 0.4 0.6 0.8 1.0

y/θ0

qU0

Ueff /U0= 0.56

θeff /θ0= 0.052

= [1+tanh(y/2θeff )]Ueff

2

Figure 24. Comparison between tanh function shown and the velocity data at x/θ0 = 0.73.

first instability. It seems illogical to use, for example, the boundary layer momentumthickness (θ0) to scale a phenomenon which does not involve the entire boundarylayer. The new scales of velocity and length will be Ueff and θeff to imply that it is theeffective velocity scale and the effective momentum thickness only that lead to thefirst instability and the formation of the first coherent motions.

A logical choice for the velocity scale is Ueff = 0.54U0. This is justified by theapparent convection velocity of the first instability shown in figure 16. The effectivemomentum thickness of the sub-shear layer was determined by fitting the hyperbolictangent profile to the mean flow measurements acquired at x/θ0 = 0.73. The data werefit to the equation

q(y)

U0

=Ueff

2

[1 + tanh

(y

2θeff

)](7)

in the range 0 < q < 0.54U0. The mean velocity data and equation (7) are ingood agreement over this range, which corresponds to |y/θ0| < 0.1; see figure 24.Equation (7) was used to find θeff because the linear stability theory calculationof the most amplified wavenumber is derived from the tanh profile. The effectivemomentum thickness was determined to be θeff =0.5 mm (θ0/θeff = 19.2) by a least-squares fit between the data points and equation (7). Finally, the values fm =130 Hz,Ueff = 0.54U0 = 3.98 m s−1, and θeff = 0.5 mm lead to St0 = 0.0163. Although theuncertainty of this measurement is estimated to be ±10%, this value is in excellentagreement with both linear theory and all existing literature for shear layers originatingfrom laminar boundary layers.

Given the above observations, a more objective scaling of the initial frequency couldbe used for turbulent separation. Specifically, the realization that only the near-wallvorticity participates in the first instability implies that the wall scaling could beused. That is, using the viscous wall units defined for the boundary layer upstreamof separation, the dimensionless frequency can be written as St+

0 = f0ν/u2τ . For the

present data, St+0 = 0.025. It remains a question of interest if this remains constant

at other turbulent Reynolds numbers. It is noted that the value computed (from theBlasius solution) for laminar separation is St+

0 = 0.075.


14 cm

y

zxU0

Figure 25. Schematic of eight-wire rake in the shear layer.

6. Experiment 4: spanwise correlation measurementsThe statistical data presented thus far were acquired at a single spanwise location.

Because the flow is stochastically homogeneous, these data are representative of theentire flow field excluding the sidewall boundary layers (z/θ0 ≈ ±90). In addition tothese data, it is of interest to document the multi-point spanwise correlation of thevelocity field. This interest originated from the velocity measurements taken on theentrainment side of the shear layer which indicated that motions that exists withinthe shear layer have sufficient coherence to be detected well into the irrotationalflow. The spanwise extent of these coherent motions is of considerable importance inapplications involving acoustics, fluid–structure interactions, etc.

The data were acquired using a rake of eight single-wire sensors aligned in thez-direction. The sensors were equally spaced at �z/θ0 = 2.05. A schematic of therake and its orientation is shown in figure 25. These probes were traversed throughthe near-separation region using the same (x, y) locations as those shown in figure 8.The time series data were recorded for 60 s (tU0/θ0 = 4.4 × 104) at a data acquisitionrate of 5000 Hz at each spatial location.

The presentation of these results is divided into the two following sub-sections. First,several short time segments of velocity from the low-speed side will be presented.These data will help to orient the reader to the data acquisition procedures andthey will assist in the interpretation of the results. Secondly, data will be providedfrom calculations of the zero-time-delay cross-correlation for the (x, y, �z) positionsmeasured.

6.1. Example realizations in the irrotational stream

Four streamwise locations were selected to provide example realizations from theeight-wire rake: x/θ0 = 3.53, 12.0, 40.5, and 84.2. The velocity fluctuations are shownas contour plots in figure 26(a–d). The abscissa represents the time elapsed from anarbitrary starting point (i.e. when the A/D system was activated) and was not intendedto isolate any specific observed features. The time was non-dimensionalized using thelocal value of the peak frequency as shown in figure 17; that is, λ≡ (time) × fm whichrepresents the data in terms of the integer number of averaged wave periods. Tenwave periods are shown in these figures. Longer time records yield qualitatively similarrepresentations of the flow field. The contours show the fluctuating component ofvelocity from all eight sensors normalized by the standard deviation of the respectivesignals. The legend for the grey scale shows that light shading indicates positive


0

2

4

6

8

10

12

14

2 4 6 8 10λ

zθ0

0123

–1–2–3

0 3.01.5–3.0 –1.5

z/θ0=2.05

z/θ0=12.4

cc=0.02

(a)

0

2

4

6

8

10

12

14

2 4 6 8 10λ

zθ0

0123

–1–2–3

0 3.01.5–3.0 –1.5

z/θ0=2.05

z/θ0=12.4

cc=0.23

(b)

0

2

4

6

8

10

12

14

2 4 6 8 10λ

zθ0

0123

–1–2–3

0 3.01.5–3.0 –1.5

z/θ0=2.05 z/θ0=12.4 cc=0.89

(d )

0

2

4

6

8

10

12

14

2 4 6 8 10λ

zθ0

0123

–1–2–3

0 3.01.5–3.0 –1.5

z/θ0=2.05z/θ0=12.4 cc=0.75

(c)

Figure 26. Contour plots show magnitude of velocity fluctuations from eight sensorsnormalized by the standard deviation of the respective velocity fluctuations. The contoursare interpolated between the eight discrete locations. The upper x, y plots show timetraces taken from wires located at z/θ0 = 2.05 and 12.4 for the same time period. Thecorrelation coefficient for these two q(t) signals from the full time record is indicated by‘cc’. (a) x/θ0 = 3.53, y/θ0 = −3.7; (b) x/θ0 = 12.0, y/θ0 = −5.37; (c) x/θ0 = 40.5, y/θ0 = −10.9;(d) x/θ0 = 84.2, y/θ0 = −19.6.

fluctuations in velocity, and dark shading negative fluctuations. For each of thecontour plots, the time traces from the wires located at z/θ0 = 2.05 and 12.4 areshown for comparison.

Growth in the spanwise coherence of the entrainment velocity was observed as x/θ0

was increased from 3.53 to 84.2. Specifically, figure 26(a) shows clear evidence that


0

0.8

0.6

0.4

–0.20 1 510

∆z/θ0

(x/θ0=3.53, y/θ0= –3.7)

5 2 0

0.2

1.0

R(∆z)

(x/θ0=12.0, y/θ0= –5.7)

(x/θ0= 40.5, y/θ0= –10.9)

(x/θ0=84.2, y/θ0= –19.6)

(x= 0, y/θ0= 0.5)

Figure 27. Spanwise correlation coefficient of the data shown in figure 26.

the motions which led to these fluctuations do exhibit a coherence in the spanwisedirection, although the spatial extent is limited. Figure 26(b) could be describedsimilarly, although the spatial extent of the coherent motions has obviously ‘grown’.It is instructive to reiterate the idea of the sub-shear layer when considering thesespanwise correlations. As a specific example, recall that at the streamwise locationx/θ0 = 12, the flow field for y/θ0 > 2 is stochastically identical to the boundary layer.Therefore, the spanwise correlation of the motions is observed to be several timeslarger than the local sub-shear layer thickness.

In order to directly contrast this large spanwise coherence with the boundary layerflow, the correlation coefficient for these four locations as well as one position withinthe boundary layer (at x = 0, y = 0.5θ0) are shown in figure 27. It is clear from thesemeasurements that the larger spanwise coherence is developed downstream of theseparation point. The correlation values are considerably larger in the low-speedregion near the sub-shear layer than in the boundary layer, even at small x/θ0

values.The continued growth in coherence is also seen in figures 26(c) and 26(d), where the

irrotational fluid motion approaches that of a two-dimensional flow. These measure-ments are similar to those of Bowand & Trout (1980, 1985). Twelve wires were placedin the low-speed side of a two-stream shear layer facility in the Browand & Troutinvestigation. These authors observed coherent velocity fluctuations in the irrotationalflow near to the active shear layer, with a correlation length scale that increased inthe streamwise direction. Also, the short time realizations of the velocity fluctuationsshown in Browand & Trout are qualitatively similar to those shown in figure 26.

6.2. Correlation coefficient data

The correlation coefficient R(�z) was calculated (as shown in figure 27) for each ofthe (x, y) locations (see figure 8 for the x, y data points used). The domain of thedata set can be represented by the ‘volume’ of points (x, y, �z). The visualization of


Figure 28. Spanwise correlation field at (a) x/θ0 = 3.53, (b) 12.0, (c) 40.6, (d) 87.2. The

contours represent the magnitude of q ′(x, y, z)q ′(x, y, z + �z)/(q′2).

these data was made possible by ‘slicing’ the volume in planes perpendicular to eitherthe x- or the z-direction. First, the streamwise locations x/θ0 = 3.53, 12.0, 40.5, and84.2 (i.e. the positions used in figure 26) were selected. Figure 28(a–d) shows contoursof the correlation coefficient at these locations. The mean velocity profile is includedwith each of the figures for reference. Figure 29(a–d) shows contours of R at specific


0

20

–20

0 5 0 100

(a)

yθ0

0

20

–20

0 5 0 100

(b)

yθ0

0

20

–20

0 5 0 100

(c)

yθ0

0

20

–20

0 5 0 100

(d )

yθ0

0

20

–20

0 5 0 100

(e)

yθ0

x/θ0

Figure 29. Correlation coefficient (as defined in figure 28) at (a) �z/θ0 = 2.08, (b) 4.16,(c) 6.24, (d) 8.32, (e) 14.5. Note that the solid boundary represents the spatial extent ofthe measurements. The dashed boundary represents the region where urms/U0 > 0.01. Themeasurement locations within this domain are identified by the grid. A schematic of theboundary layer mean profile near x = y =0 is shown for reference.


�z values. The solid boundary surrounding the contour plot represents the spatialextent of the data acquisition. The dashed boundary represents the contour line wherethe fluctuation levels are 1% of the free-stream velocity (see figure 11). Correlationdata are shown in the region where the standard deviation of the fluctuations exceeds1% of U0.

Several important features of the flow field can be observed from thesemeasurements. First, the central region of the shear layer shows quite low spanwisecorrelation values. This is in contrast to the low-speed irrotational flow measurementswhich exhibit a large correlation length with respect to the local length scales.Although it is apparent that the irrotational fluctuations are caused by the motionsexisting in the turbulent fluid, the correlation magnitudes are diminished by thepresence of relatively large fluctuating velocities that can be attributed to small-length-scale motions.

A second feature observed from these data is the difference between the high- andlow-speed sides of the shear layer. Specifically, the correlation length on the low-speedside grows smoothly and continuously from the separation point, whereas the high-speed side shows an abrupt chage in correlation values near x/θ0 = 55. These datareinforce the inferred presence of the sub-shear layer, and they also provide insightinto how the sub-shear layer evolves into a fully developed shear layer. Specifically,the vortical fluid which occupied the outer region of the boundary layer does notparticipate in the instability which leads to the large-scale spanwise motions. Atincreasing streamwise positions, additional vortical fluid is drawn in to ‘participate’with the coherent motions. The dramatic change in the correlation field near x/θ0 ≈ 55indicates that the motions have become strong enough to have a measurable effect onthe high-speed irrotational fluid. It is instructive that the time-averaged convectionvelocity of the coherent motions at this location is approximately 82% of the finalvalue of 0.5U0 (see figure 16).

7. Conclusions with discussionThe experimental data described in the previous sections lead to a number of

observations about the transition from an attached boundary layer to a single-streamshear layer. The significant difference between the present study and previous researchis the moderately high Reynolds number at separation (Reθ = 4650), and a relativelylarge integral length scale (θ0 = 9.6 mm) which allowed excellent spatial resolutionof the measurements. In this section each main conclusion is listed, followed by adiscussion and interpretation of the relevant data sets.

(i) The stochastic properties of the boundary layer at separation (x = 0) appear to beunaffected by the flow field downstream of separation.

This conclusion is supported by several of the experiments. First, the zero cross-correlation magnitudes between velocities measured in the boundary layer and inthe entrainment stream is a strong indication that a coupling does not exist betweenthe two regions. The lack of correlation in the low-pass-filtered time series furthersupports this result. The reported stochastic properties of the boundary layer showthat the moments of the velocity and vorticity fields are in good agreement with theexisting literature on flat-plate boundary layers without separation. If the shear layerdid have an effect on the boundary layer at separation, it was not apparent in any ofthe point statistics or correlation measurements.

The streamwise invariance of the statistics in the outer region of the boundary layerprovides additional evidence to support this conclusion. In other words, not only do


the shear layer fluctuations not affect the boundary layer at x =0, there is also nomeasurable effect on much of the y > 0 region for a considerable length downstream.This will be discussed further in (ii) below.

The fact that a high-Reynolds-number boundary layer which separates at a 90◦ edgeis not affected by the separation is important for several reasons. If the separationregion were to be modelled using a computational method, the upstream boundaryconditions could be specified at x = 0 by the stochastic properties of the approachboundary layer. If this conclusion were not supported by the observations, thenthe separated flow field would have been found to affect the stochastic propertiesupstream of separation, and the inlet conditions to the computation would have tobe specified at some x < 0, rather than at x = 0.

It is likely that communication from the boundary layer to the shear layer isdependent on the Reynolds number. Laminar and low-Reynolds-number turbulentflows at separation are likely to be more sensitive to the perturbations caused bythe downstream motions. A future study of this topic could attempt to measure anobjective indication of the communication for a range of Reynolds numbers.

(ii) The outer region of the boundary layer is stochastically streamwise invariant forseveral integral lengths downstream of separation.

This streamwise invariance of the boundary layer properties is evident in all ofthe data identified above. For example, the plots showing statistics in figures 8–11and 29 all show invariant properties of the flow in the region 0< x/θ0 < 50 at lateralpositions of y > 0.15x. This adds information to a body of literature which existson the topic of inner and outer scales in the boundary layer; see, for exampleKlewicki (1989), Marusic & Perry (1990), and Hamelin & Alving (1996). Thepresent observations indicate that not only are the outer scales not dependenton the local production of near-wall turbulence, but that the flow is not evenaffected by the large-scale fluctuations of the shear layer measured in the y ≈ 0region.

(iii) The separation region can be described as a ‘sub-shear layer’ in which only thenear-wall vorticity participates in the initial shear layer instability.

The region which defines the sub-shear layer is qualitatively evident in the flowvisualization image shown in figure 6. This may also be identified as the wedge-shapedregion roughly defined by urms/U0 > 0.087 on figure 11. Many of the salient featuresthat are associated with canonical shear layers have been identified in this region.For example, it can also be observed that the ‘spread angle’ of the sub-shear layer isequal to that of the fully developed shear layer. This can be quantified from the twocontour lines for which urms/U0 = 0.087 on figure 11. The normalized distance betweenthese lines, say, �y, can be expressed as �y/x. This was found to be 0.17 ± 0.01 for5 < x/θ0 < 650. That is, the spread angle is the same in the sub-shear layer and thefully developed shear layer.

An important feature of shear layers is the presence of coherent motions whichdevelop from the inflectional instability mechanism (see Ho & Huerre 1984 fora review). These motions are known to move in the streamwise direction with awell-defined convection velocity. In fully developed single-stream shear layers, thisvelocity is known to be nominally 2Uc/U0 = 1.0. The convection velocity in thesub-shear layer region is shown in figure 16. These data indicate a dramatic risein the convection velocity from 2Uc/U0 ≈ 0.54 at separation to a value near 1.0 atx/θ0 = 120. It is inferred from these data that only the near-wall vorticity is ‘rollingup’ or participating in the inflectional instability. A quantitative estimate of thefluid domain which participates in the initial roll-up has been identified as that


below the q/U0 = 0.54 isotach. Equivalently, this bound is given by y+ < 42 for thepresent (Reθ = 4650) boundary layer. As the sub-shear layer grows into the outerpart of the boundary layer, more of the vorticity is incorporated into the coherentmotions thereby increasing both the width of the sub-shear layer and the convectionvelocity.

The present understanding of the vorticity field within a turbulent boundary layeris consistent with the current findings. That is, the stochastic and instantaneousproperties of the motions which define the inner and outer scales of the boundarylayer are significantly different. The near-wall motions are described by, for exampleAdrian, Meinhart & Tomkins (2000), and Panton (2001). In these references, thenear-wall region is understood to be composed of groups of coherent motions whichdefine localized shear layers. These result in ‘vortex packets’ which evolve into theouter scales of motion. It is intuitively reasonable that the near-wall region witha high level of organization of vorticity would be subject to inflectional instabilityimmediately downstream of separation. In contrast, the motions farther from thewall are usually considered to be more disorganized and not closely correlated withthe motions very near to the wall. The current measurements of spanwise vorticityfluctuations at x = 0 support this viewpoint. It is therefore reasonable that these outermotions would not participate in the roll-up phenomenon.

(iv) The recognition of the sub-shear layer allows a physical explanation for whyshear layers with a turbulent boundary layer at separation approach self-similarity atsmaller x/θ0 values than shear layers with a laminar separation.

The streamwise location where the flow field becomes self-similar has been found tobe dependent on the state and Reynolds number of the boundary layer at separation.For example, Bradshaw (1966) recommends a value of x/θ0 > 1000 for self-similaritygiven a laminar separation. Hussain & Zaman (1985), Bruns (1990), and the presentstudy have found that x/θ0 > 100 is sufficient for the mean velocity to exhibit self-preservation when the separating boundary layer is turbulent.

These differences can be explained in terms of the length scales of the turbulentvs. laminar boundary layers. Dimotakis & Brown (1976) suggested that it is not thedownstream location as defined by a number of original integral length scales (e.g.x/θ0) that is important for the establishment of a self-similar condition. Rather, it isthe number of ‘interactions’ between the large-scale motions of the flow. The numberof interaction will depend on the ratio of the streamwise location to the spacingof the initial disturbance. Specifically, the number of interactions m(x) is estimated(equation (14) of Dimotakis and Brown) to be

m(x) = log2(x/l0) (8)

where l0 is the spacing of the initial disturbance. For a laminar boundary layer atseparation, the criterion x/θ0 > 1000 is equivalent to a value of m ≈ 4. In the presentstudy the stream-wise location x/θ0 = 100 corresponds to a value of m between 6 and7, given that the spacing of the initial disturbance was l0 ≈ θ0. This small value ofinitial disturbance length compared with θ0 is a direct result of the presence of thesub-shear layer.

(v) The dimensionless frequency of the initial instability agrees with linear stabilitytheory when appropriate effective velocity and length scales are identified.

The measurement and prediction of the dimensionless frequency of the initialstability has received considerable attention in the literature; see, for example, Ho &Huerre (1984) and Thomas (1991). This focus is a result of the universal nature ofthe initial instability in a variety of shear layer and jet flow fields. Additionally,


linear theory has been quite successful in the prediction of this dimensionlessfrequency for many experiments with laminar boundary layers at separation. However,differences in the measured frequencies have been found in shear layers with turbulentboundary layers at separation. The present study has shown that these differencesare a result of the different length and velocity scales that exist in a turbulentboundary layer. Specifically, a laminar boundary layer can be fully characterizedby the free-stream velocity and momentum thickness. In contrast, a turbulentboundary layer has outer scales (free-stream and momentum thickness) and innerscales (uτ and ν/uτ ). The data presented suggest that only the near-wall vorticalfluid participates in the initial instability at this Reynolds number. It is then quitereasonable that the outer scales would not be functionally related to the measuredfrequencies.

(vi) Preferred frequency and spanwise correlation measurements have shown that thecharacteristic large-scale motions observed in self-similar single- and two-stream shearlayers are present in the sub-shear layer region.

Several investigators have studied the properties and importance of the largestscales of motion in shear layers. These motions appear to have a large spanwisecoherence and a significant fraction of the total turbulent kinetic energy of the flow.Experimentalists have sought to measure these features in order to better understandtheir role and dynamic significance; see, for example, Hussain & Zaman (1985),Browand & Weidman (1976), and Browand & Trout (1980, 1985).

The present experiments have found that a number of the features measured inthe self-similar regions of both two-stream and single-stream shear layers have alsobeen observed in the sub-shear layer region. For example, the preferred frequencyexhibited by both the sub-shear layer and self-similar shear layer can be described bythe relatively simple equation: fm ∼ x−0.71 (for x/θ0 > 1) as shown in figure 19. It canbe inferred that the physical mechanisms which lead to this preferred frequency arepresent throughout the flow field downstream of separation. Note that the exponent−0.71 is anomalous, in that the expected value from self-similarity is −1. The reasonfor this is not clear, although Hussain & Zaman (1985) show that an exponent of −1is not observed until downstream distances of x/θ0 > 300.

A second feature observed in the sub-shear layer region was the large spanwisecorrelation values observed in the near-separation region. Specifically, figure 26(a)shows evidence of spatial organization at the location of the first instability (x/θ0 ≈ 1).The correlations then grow in spanwise extent in the streamwise direction. Theseobservations are consistent with observations made by Browand & Trout (1980, 1985)in the self-similar region of a two-stream shear layer.

The observations noted above justify the terminology ‘sub-shear layer’. Specifically,the region exhibits characteristics which are associated with self-similar shear layers,while the prefix ‘sub’ denotes that the shear layer is growing adjacent to what appearsto be the outer region of a flat-plate boundary layer.

The single-stream shear layer facility was financially supported by the FordMotor Company, and primarily designed and fabricated by the first author. Thedonation of the four entrainment fans from Aerotech, Inc. of Mason, MI is gratefullyacknowledged. Partial support for a Graduate Research Assistantship to S. C. M. wasalso provided by the Office of Naval Research, grant number N00014-00-1-0753 tothe University of Utah with MSU as a sub-contractor. The grant monitor was Dr L.Patrick Purtell.


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